Quantum scattering - 1/r^2

In summary, the conversation discusses using partial waves method to calculate scattering amplitude for a quantum particle scattering off a potential of the form V(r) = a/r^2. The solution requires solving Schrodinger's equation for the specified potential, but the challenge lies in calculating phase shifts for each angular momentum's value. The question also addresses the use of Bessel spherical functions and the impact of the centrifugal term on the solutions. Finally, it is mentioned that the asymptotic properties of Bessel functions apply regardless of whether the index is an integer or not.
  • #1
neworder1
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Homework Statement



Use partial waves method to calculate scattering amplitude for a quantum particle scattering off the potential V(r) = a/r^2.

Homework Equations





The Attempt at a Solution



To calculate phase shifts [tex]\delta_{l}[/tex] for each angular momentum's value l, it's necessary to solve Schrodnger's equation for the specified potential - but I'm unable to do it. I know that solution for a free particle can be expressed in terms of Bessel spherical functions, but in radial Schrodinger's equation we have the centrifugal term of the form l(l+1)/r^2, where l is an integer, not for arbitary potential a/r^2. Any hints?
 
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  • #2
The radial equation doesn't care whether or not l is an integer ...
 
  • #3
Ok, but in the noninteger case the solutions are special functions, from which I don\t know how to calculate the phase factors in question.
 
  • #4
The asymptotic properties of Bessel functions hold whether or not the index is an integer.
 

1. What is "Quantum scattering - 1/r^2"?

"Quantum scattering - 1/r^2" refers to the mathematical formula used to describe how particles interact with one another at a distance. It is based on the inverse square law, which states that the force between two particles is inversely proportional to the square of the distance between them.

2. How does "Quantum scattering - 1/r^2" differ from classical scattering?

Classical scattering uses the laws of classical mechanics to describe the interaction of particles, while "Quantum scattering - 1/r^2" takes into account the principles of quantum mechanics. In classical scattering, the force between particles decreases linearly with distance, while in "Quantum scattering - 1/r^2", it decreases exponentially.

3. What is the significance of the 1/r^2 term in "Quantum scattering - 1/r^2"?

The 1/r^2 term represents the inverse square law, which is a fundamental principle in physics. It describes the relationship between the force of interaction between two particles and their distance apart. In "Quantum scattering - 1/r^2", this term is used to calculate the probability of a particle being scattered at a particular angle.

4. How is "Quantum scattering - 1/r^2" related to quantum tunneling?

"Quantum scattering - 1/r^2" is often used to describe the phenomenon of quantum tunneling, which is when a particle is able to pass through a potential barrier that would be impossible to overcome according to classical physics. This is because the 1/r^2 term takes into account the wave-like nature of particles, allowing for the possibility of them "tunneling" through the barrier.

5. What are some real-world applications of "Quantum scattering - 1/r^2"?

"Quantum scattering - 1/r^2" has many applications in fields such as nuclear physics, astrophysics, and materials science. It is used to study the interactions between particles in nuclear reactions, the distribution of particles in the universe, and the behavior of particles in different materials. It also has applications in developing advanced technologies, such as quantum computers and quantum communication systems.

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